Problem: Real numbers $x$ and $y$ satisfy the equation $x^2 + y^2 = 10x - 6y - 34$.  What is $x+y$?
Explanation: If we complete the square after bringing the $x$ and $y$ terms to the other side, we get \[(x-5)^2 + (y+3)^2 = 0.\]Squares of real numbers are nonnegative, so we need both $(x-5)^2$ and $(y+3)^2$ to be $0.$  This only happens when $x = 5$ and $y = -3.$ Thus, $x+y = 5 + (-3) = \boxed{2}.$